Conference Agenda

A multi-type neutral Cannings population model with mutation and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type coalescent with mutation allowing for simultaneous multiple collisions of ancestral lineages. The limiting coalescent shares the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions into reproductive and mutational parts.

We consider a general exchangeable diploid population model (Cannings model) as a model for a random pedigree. Embedded within this pedigree are the genealogies associated with a single, neutral autosomal locus of which each individual carries two copies. Mathematically, they form a system of coalescing random walks in a random environment, given by the pedigree.

Complementing previous work on an `annealed' limit theorem which states that, under mild conditions, the genealogies can asymptotically be described by a $\Xi$-coalescent after averaging over the pedigree, we establish the corresponding `quenched' limit. That is, we fix a realisation of the random pedigree and show that, in the limit of large populations, the genealogies can be described by an \emph{inhomogeneous} coalescent process.

To account for continuous features among individuals such as age or place of residence, we extend multitype Reed–Frost models, i.e., discrete-time SIR models, such that the distribution of certain features may be modeled as a probability distribution on a suitable space of possible types of individuals. The epidemic then runs on a population randomly drawn from this distribution while the infection and recovery probabilities may also depend on the individuals’ types. Our main results state that for every point in time, the empirical measures of susceptible, infective and recovered individuals converge pointwise in probability to a deterministic limit given by recursions (resembling those for Reed–Frost epidemics) and that those measures fulfill a central limit theorem. We discuss our model assumptions and illustrate the results with simulations.